Size effects in the toughening of brittle materials by heterogeneities: a non-linear analysis of front deformations

Abstract

Traditional computational approaches in simulating crack propagation in perfectly brittle materials rely on the estimate of stress intensity factors along the rupture front. This proves highly challenging in 3D when the crack geometry departs from very specific cases for which analytical solutions are available, like e.g. the penny-shaped crack geometry. Here, we extend the first-order theory of Gao and Rice (1987), and predict the distribution of the mode I stress intensity factor $K_\mathrm{I}$ along the front of a tensile coplanar crack that is slightly perturbed from a reference penny-shaped configuration, up to second order in the perturbation amplitude. Our theory is validated against analytical solutions available for embedded elliptical cracks, and its range of validity is further assessed using numerical simulations performed on cosine front perturbations of varying mode and amplitude. It is then used to develop a homogenization framework for the toughness of weakly disordered media. The effective toughness and its fluctuations are bridged quantitatively to the intensity of the toughness fluctuations and their spatial structure. Our theoretical predictions are compared to the results of ~1 million simulations of crack propagation building on our second-order theory and Fast Fourier Transforms. We show that the impact of toughness heterogeneities is size-dependent, as they generally weaken the material when the crack size is lower or comparable to the typical heterogeneity size, but reinforces it otherwise. It results in an apparent R-curve behavior of the brittle composite at the macroscale.